Noetherian group: Difference between revisions

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{{group property}}
{{group property}}
{{variationof|finiteness (groups)}}
{{variationof|finiteness (groups)}}
 
[[importance rank::3| ]]
==Definition==
==Definition==


===Symbol-free definition===
===Symbol-free definition===


A [[group]] is said to be '''slender''' or '''Noetherian''' or to satisfy the '''maximum condition on subgroups''' if it satisfies the following equivalent conditions:
A [[group]] is said to be '''Noetherian''' or '''slender''' or to satisfy the '''maximum condition on subgroups''' if it satisfies the following equivalent conditions:


# Every [[subgroup]] is [[finitely generated group|finitely generated]]
# Every [[subgroup]] is [[finitely generated group|finitely generated]]
# Any [[ascending chain of subgroups]] stabilizes after a finite length
# Any ascending chain of subgroups stabilizes after a finite length.
# Any nonempty collection of subgroups has a maximal element: a member of that collection that is not contained in any other member of the collection.
# Any nonempty collection of subgroups has a maximal element: a member of that collection that is not contained in any other member of the collection.


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{{obtainedbyapplyingthe|hereditarily operator|finitely generated group}}
{{obtainedbyapplyingthe|hereditarily operator|finitely generated group}}
==Metaproperties==
{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[Noetherianness is subgroup-closed]] || If <math>G</math> is a Noetherian group and <math>H \le G</math> is a subgroup, then <math>H</math> is Noetherian.
|-
| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[Noetherianness is quotient-closed]] ||If <math>G</math> is a Noetherian group and <math>H \le G</math> is a [[normal subgroup]], the [[quotient group]] <math>G/H</math> is Noetherian.
|-
| [[satisfies metaproperty::extension-closed group property]] || Yes || [[Noetherianness is extension-closed]] || If <math>H</math> is a normal subgroup of <math>G</math> such that both <math>H</math> and <math>G/H</math> are Noetherian, then <math>G</math> is also Noetherian.
|}


==Relation with other properties==
==Relation with other properties==
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
|-
| [[Weaker than::Finite group]] || || || || {{intermediate notions short|slender group|finite group}
| [[Weaker than::finite group]] || || || || {{intermediate notions short|slender group|finite group}}
|-
|-
| [[Weaker than::Finitely generated abelian group]] || [[finitely generated group|finitely generated]] and [[abelian group|abelian]] || || || {{intermediate notions short|slender group|finitely generated abelian group}}
| [[Weaker than::finitely generated abelian group]] || [[finitely generated group|finitely generated]] and [[abelian group|abelian]] || || || {{intermediate notions short|slender group|finitely generated abelian group}}
|-
|-
| [[Weaker than::Finitely generated nilpotent group]] || [[finitely generated group|finitely generated]] and [[nilpotent group|nilpotent]] || || || {{intermediate notions short|slender group|finitely generated nilpotent group}}
| [[Weaker than::finitely generated nilpotent group]] || [[finitely generated group|finitely generated]] and [[nilpotent group|nilpotent]] || || || {{intermediate notions short|slender group|finitely generated nilpotent group}}
|-
|-
| [[Weaker than::Supersolvable group]] || has a [[normal series]] where all successive quotients are cyclic || || || {{intermediate notions short|slender group|supersolvable group}}
| [[Weaker than::supersolvable group]] || has a [[normal series]] where all successive quotients are cyclic || || || {{intermediate notions short|slender group|supersolvable group}}
|-
|-
| [[Weaker than::Polycyclic group]] || slender and [[solvable group|solvable]] || || || {{intermediate notions short|slender group|polycyclic group}}
| [[Weaker than::polycyclic group]] || slender and [[solvable group|solvable]] || || || {{intermediate notions short|slender group|polycyclic group}}
|}
|}


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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
|-
| [[Stronger than::Finitely generated group]] || || || || {{intermediate notions short|finitely generated group|slender group}}
| [[Stronger than::Finitely generated group]] || || || || {{intermediate notions short|finitely generated group|Noetherian group}}
|-
|-
| [[Stronger than::Maximal-covering group]] || || || || {{intermediate notions short|maximal-covering group|slender group}}
| [[Stronger than::Maximal-covering group]] || || || || {{intermediate notions short|maximal-covering group|Noetherian group}}
|-
|-
| [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]] || || || || {{intermediate notions short|group satisfying ascending chain condition on subnormal subgroups|slender group}}
| [[Stronger than::Group satisfying ascending chain condition on subnormal subgroups]] || || || || {{intermediate notions short|group satisfying ascending chain condition on subnormal subgroups|Noetherian group}}
|-
|-
| [[Stronger than::Group satisfying subnormal join property]] || || || || {{intermediate notions short|group satisfying subnormal join property|slender group}}
| [[Stronger than::Group satisfying subnormal join property]] || || || || {{intermediate notions short|group satisfying subnormal join property|Noetherian group}}
|-
|-
| [[Stronger than::Group satisfying ascending chain condition on normal subgroups]] || || || || {{intermediate notions short|group satisfying ascending chain condition on subnormal subgroups|slender group}}
| [[Stronger than::Group satisfying ascending chain condition on normal subgroups]] || || || || {{intermediate notions short|group satisfying ascending chain condition on subnormal subgroups|Noetherian group}}
|-
|-
| [[Stronger than::Hopfian group]] || every [[surjective endomorphism]] is an [[automorphism]] || [[Slender implies Hopfian]]||[[Hopfian not implies slender]] || {{intermediate notions short|Hopfian group|slender group}}
| [[Stronger than::Hopfian group]] || every [[surjective endomorphism]] is an [[automorphism]] || [[Noetherian implies Hopfian]]||[[Hopfian not implies Noetherian]] || {{intermediate notions short|Hopfian group|Noetherian group}}
|-
|-
| [[Stronger than::Finitely generated Hopfian group]] || finitely generated and Hopfian || [[Slender implies Hopfian]]|||| {{intermediate notions short|finitely generated Hopfian group|slender group}}
| [[Stronger than::Finitely generated Hopfian group]] || finitely generated and Hopfian || [[Noetherian implies Hopfian]]|||| {{intermediate notions short|finitely generated Hopfian group|Noetherian group}}
|-
| [[Stronger than::Minimax group]] || || || || {{intermediate notions short|minimax group|slender group}}
|-
| [[Stronger than::Group in which every locally finite subgroup is finite]] || || || || {{intermediate notions short|group in which every locally finite group is finite|Noetherian group}}
|}
|}


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|-
|-
| [[Weaker than::Polycyclic group]] || [[solvable group]] || {{intermediate notions short|slender group|polycyclic group}} || {{intermediate notions short|polycyclic group|solvable group}}
| [[Weaker than::Polycyclic group]] || [[solvable group]] || {{intermediate notions short|slender group|polycyclic group}} || {{intermediate notions short|polycyclic group|solvable group}}
|}
===Related properties===
{| class="sortable" border="1"
! Property !! Meaning !! Proof of one non-implication !! Proof of other non-implication !! Notions stronger than both !! Notions weaker than both
|-
| [[Artinian group]] || any descending chain of subgroups stabilizes after finitely many steps || [[Noetherian not implies Artinian]] || [[Artinian not implies Noetherian]] || {{stronger than both short|Noetherian group|Artinian group}} || {{weaker than both short|Noetherian group|Artinian group}}
|-
| [[Finitely presented group]] || has a finite [[presentation of a group|presentation]] || [[Noetherian not implies finitely presented]] || [[finitely presented not implies Noetherian]] || {{stronger than both short|Noetherian group|finitely presented group}} || {{weaker than both short|Noetherian group|finitely presented group}}
|}
|}

Latest revision as of 06:28, 7 June 2012

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

Definition

Symbol-free definition

A group is said to be Noetherian or slender or to satisfy the maximum condition on subgroups if it satisfies the following equivalent conditions:

  1. Every subgroup is finitely generated
  2. Any ascending chain of subgroups stabilizes after a finite length.
  3. Any nonempty collection of subgroups has a maximal element: a member of that collection that is not contained in any other member of the collection.

Formalisms

In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: finitely generated group
View other properties obtained by applying the hereditarily operator

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes Noetherianness is subgroup-closed If is a Noetherian group and is a subgroup, then is Noetherian.
quotient-closed group property Yes Noetherianness is quotient-closed If is a Noetherian group and is a normal subgroup, the quotient group is Noetherian.
extension-closed group property Yes Noetherianness is extension-closed If is a normal subgroup of such that both and are Noetherian, then is also Noetherian.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite group |FULL LIST, MORE INFO
finitely generated abelian group finitely generated and abelian |FULL LIST, MORE INFO
finitely generated nilpotent group finitely generated and nilpotent |FULL LIST, MORE INFO
supersolvable group has a normal series where all successive quotients are cyclic |FULL LIST, MORE INFO
polycyclic group slender and solvable |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finitely generated group |FULL LIST, MORE INFO
Maximal-covering group |FULL LIST, MORE INFO
Group satisfying ascending chain condition on subnormal subgroups |FULL LIST, MORE INFO
Group satisfying subnormal join property |FULL LIST, MORE INFO
Group satisfying ascending chain condition on normal subgroups |FULL LIST, MORE INFO
Hopfian group every surjective endomorphism is an automorphism Noetherian implies Hopfian Hopfian not implies Noetherian |FULL LIST, MORE INFO
Finitely generated Hopfian group finitely generated and Hopfian Noetherian implies Hopfian |FULL LIST, MORE INFO
Minimax group |FULL LIST, MORE INFO
Group in which every locally finite subgroup is finite |FULL LIST, MORE INFO

Conjunction with other properties

Conjunction Other component of conjunction Intermediate notions between slender group and conjunction Intermediate notions between other component and conjunction
Finitely generated abelian group abelian group |FULL LIST, MORE INFO |FULL LIST, MORE INFO
Finitely generated nilpotent group nilpotent group |FULL LIST, MORE INFO |FULL LIST, MORE INFO
Polycyclic group solvable group |FULL LIST, MORE INFO |FULL LIST, MORE INFO

Related properties

Property Meaning Proof of one non-implication Proof of other non-implication Notions stronger than both Notions weaker than both
Artinian group any descending chain of subgroups stabilizes after finitely many steps Noetherian not implies Artinian Artinian not implies Noetherian |FULL LIST, MORE INFO |FULL LIST, MORE INFO
Finitely presented group has a finite presentation Noetherian not implies finitely presented finitely presented not implies Noetherian |FULL LIST, MORE INFO |FULL LIST, MORE INFO
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